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Bui Xuan Hai

Publications and source records attributed to Bui Xuan Hai.

At least 19 recordsLinked to original sources
arXiv2026

On Locally Generalized radical linear groups over rings

In this paper, we investigate the structure of locally generalized radical linear groups over certain non-commutative rings and algebras. The base rings and algebras we consider include division rings, left artinian rings, locally finite algebras over fields, and PI-algebras over fields. Among various results, we get the positive answers to the General Burnside Problem and to Baer's Conjecture for some particular cases of linear groups.

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arXiv2025

On free subgroups in Leavitt path algebras

Let $E$ be a graph and $K$ a field. In this paper we prove that the multiplicative group of a unital noncommutative Leavitt path algebra $L_K(E)$ contains non-cyclic free subgroups provided $K$ is of characteristic $0$. Further, we provide a description of the generators of such free subgroups in term of the graph $E$.

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arXiv2024

Subnormal subgroups of almost locally simple artinian algebras with involutions

In this paper, we investigate subnormal subgroups of the multiplicative group of an almost locally simple artinian algebra with involution. In particular, we show that if either the set of traces or the set of norms of such a subgroup with respect to this involution is central, then the algebra must be either a quaternion division algebra or the matrix ring of degree $2$ over a field.

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arXiv2023

Multiplicative groups of Leavitt path algebras

In this paper, we prove that the multiplicative group of a unital non-commutative Leavitt path algebra $L_K(E)$ and Cohn path algebra $C_K(E)$ contain a non-cyclic free subgroup, provided $K$ is a non-absolute field. We also provide a description of the generators of free subgroups in term of the graph $E$. Finally, we determine multiplicative groups of Leavitt path algebras of some special types.

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arXiv2022

Locally solvable maximal subgroups in division rings

Let $D$ be a division ring with center $F$, and $G$ an almost subnormal subgroup of $D^*$. In this paper, we show that if $G$ contains a non-abelian locally solvable maximal subgroup, then $D$ must be a cyclic algebra of prime degree over $F$. Moreover, it is proved that every locally nilpotent maximal subgroup of $G$ is abelian.

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arXiv2020

Multiplicative Subgroups in weakly locally finite division rings

The description of the subgroup structure of a non-commutative division ring is the subject of the intensive study in the theory of division rings in particular, and of the theory of skew linear groups in general. This study is still so far to be complete. In this paper, we study this problem for weakly locally finite division rings. Such division rings constitute a large class which strictly contains the class of locally finite division rings.

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arXiv2020

Algebraic commutators with respect to subnormal subgroups in division rings

Let $D$ be a division ring and $K$ a subfield of $D$ which is not necessarily contained in the center $F$ of $D$. In this paper, we study the structure of $D$ under the condition of left algebraicity of certain subsets of $D$ over $K$. Among results, it is proved that if $D^*$ contains a noncentral normal subgroup which is left algebraic over $K$ of bounded degree $d$, then $[D:F]\le d^2$. In case $K=F$, the obtained results show that if either all additive commutators or all multiplicative commutators with respect to a noncentral subnormal subgroup of $D^*$ are algebraic of bounded degree $d$ over $F$, then $[D:F]\le d^2$.

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arXiv2020

On locally finite skew group algebras

In this note, we study the problem on the existence of non-cyclic free subgroups of the skew group algebra of a locally finite group over a field.

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arXiv2020

Locally solvable and solvable-by-finite maximal subgroups of $GL_n(D)$

This paper aims at studying solvable-by-finite and locally solvable maximal subgroups of an almost subnormal subgroup of the general skew linear group $\GL_n(D)$ over a division ring $D$. It turns out that in the case where $D$ is non-commutative, if such maximal subgroups exist, then either it is abelian or $[D:F]<\infty$. Also, if $F$ is an infinite field and $n\geq 5$, then every locally solvable maximal subgroup of a normal subgroup of $\GL_n(F)$ is abelian.

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arXiv2020

Intersection graphs of almost subnormal subgroups in general skew linear groups

Let $D$ be a division ring, $n$ a positive integer, and GL$_n(D)$ the general linear group of degree $n$ over $D$. In this paper, we study the induced subgraph of the intersection graph of GL$_n(D)$ generated by all non-trivial proper almost subnormal subgroups of GL$_n(D)$. We show that this subgraph is complete if it is non-null. This property will be used to study subgroup structure of a division ring. In particular, we prove that every non-central almost subnormal subgroup of the multiplicative group $D^*$ of a division ring $D$ contains a non-central subnormal subgroup of $D^*$.

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arXiv2019

A note On subgroups in a division ring that are left algebraic over a division subring

Let $D$ be a division ring with center $F$ and $K$ a division subring of $D$. In this paper, we show that a non-central normal subgroup $N$ of the multiplicative group $D^*$ is left algebraic over $K$ if and only if so is $D$ provided $F$ is uncountable and contained in $K$. Also, if $K$ is a field and the $n$-th derived subgroup $D^{(n)}$ of $D^{*}$ is left algebraic of bounded degree $d$ over $K$, then $\dim_FD\le d^2$.

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arXiv2019

On perfect order subsets in finite groups

If $G$ is a finite group and $x\in G$ then the set of all elements of $G$ having the same order as $x$ is called {\em an order subset of $G$ determined by $x$} (see [2]). We say that $G$ is a {\em group with perfect order subsets} or briefly, $G$ is a {\em $POS$-group} if the number of elements in each order subset of $G$ is a divisor of $|G|$. In this paper we prove that for any $n\geq 4$, the symmetric group $S_n$ is not $POS$-group. This gives the positive answer to one of two questions rising from Conjecture 5.2 in [3].

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